Given $ m \angle RPS = 5x - 17$, $ m \angle QPR = 5x - 23$, and $ m \angle QPS = 110$, find $m\angle RPS$. $P$ $Q$ $S$ $R$
Solution: From the diagram, we see that together ${\angle QPR}$ and ${\angle RPS}$ form ${\angle QPS}$ , so $ {m\angle QPR} + {m\angle RPS} = {m\angle QPS}$ Substitute in the expressions that were given for each measure: $ {5x - 23} + {5x - 17} = {110}$ Combine like terms: $ 10x - 40 = 110$ Add $40$ to both sides: $ 10x = 150$ Divide both sides by $10$ to find $x$ $ x = 15$ Substitute $15$ for $x$ in the expression that was given for $m\angle RPS$ $ m\angle RPS = 5({15}) - 17$ Simplify: $ {m\angle RPS = 75 - 17}$ So ${m\angle RPS = 58}$.